Temporal Logics on Strings with Prefix Relation
Résumé
We show that linear-time temporal logic over concrete domains made of finite strings and the prefix relation admits a PSpace-complete satisfiability problem. Actually, we extend a known result with the concrete domain made of the set of natural numbers and the greater than relation (corresponding to the singleton alphabet case) and we solve an open problem mentioned in several publications. Since the prefix relation is not a total ordering, it is not possible to take advantage of existing techniques dedicated to temporal logics with concrete domains that are essentially linearly ordered structures. Instead, we introduce an adequate encoding of string constraints into length constraints that allows us to reduce the problem on strings to the problem on natural numbers. To do so, we also propose an extended version of the logic on strings that is able to compare lengths of longest common prefixes and for which the satisfiability problem is shown in PSpace. Finally, we show how to lift the result for the branching-time case in order to get decidability when the underlying temporal logic is CTL*.
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